Figure 1

Relaxation of a single displaced phononic mode, characterized by $f_R(x)=A\cos (k_{0}x)$. Solid yellow lines are obtained by dynamical refermionization, i.e., Eqs. (17)–(19), (21) in the thermodynamic limit $L\to \infty$. Top: Bethe-fermions occupations in the vicinity of the Fermi sea right-border. Bottom: energy in each phononic mode after relaxation, ${\epsilon }_k=|k|v⟨A_{R,n}^{†}{A}_{R,n}{⟩}_{\infty }$, where $k=2n\pi /L$ (respectively, $-2n\pi /L$) for right movers (respectively, left movers). The dash-dotted black lines are the results expected if the system were relaxing to thermal equilibrium. Dashed line shows the distribution ${\epsilon }_k^{r,\mathrm{th}}$ corresponding to thermal redistribution of the initial energy within the right-mover phonons only and the insets highlight its difference with the dynamical refermionization results. Crosses are exact numerical results obtained in the limiting case of hard-core bosons, performed for 100 atoms (see text), using $k_0=5\times 2\pi /L$ in the left figure (respectively, $k_0=2\times 2\pi /L$ in the central figure) for $A=1.0$ (respectively, $A=3.0$), and the relaxed phonon population is computed in the fermionic diagonal ensemble. Those numerical results are performed for a lattice gas of 1000 sites, which represents faithfully the continuous gas only for small $k$.